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```					STAGE 2 MATHEMATICAL APPLICATIONS                                               NAME: POSSIBLE SOLUTION

DIRECTED INVESTIGATION: COSTING CALCULATIONS

ONE POSSIBLE SOLUTION
Introduction
Christopher works part time in his parents fruit and vegetable shop, and wishes to become more involved in the business and
investigates the feasibility of selling gift baskets, containing fresh fruit and other products. By considering fixed and variable costs,
and taxation implications, he plans to calculate the number of gift baskets to be sold in order to break-even, and to consider an
appropriate sale price to determine the potential profit and therefore maximum income for himself for the first year of operation.

Mathematical Investigations, Analysis and Discussions

1.        Fixed costs may include: a portion of electricity, telephone, rental.
Variable costs may include: contents of the baskets (fruit, dried fruit, nuts, confectionary) and the basket and trimmings (ribbons,
cellophane etc)

2.       Sale Price \$30.00
Marginal Income = SP – VP
\$30 - \$18
\$12
Breakeven number = fixed costs/ MP
=3000/12
= 250

Revenue = 250 x \$30 = \$7500

(Similar calculations for Sale Price of \$40 and \$50 should be shown)

Sale Price             Variable Costs            Fixed Costs           Breakeven No.           Revenue
\$30                      \$18                     \$3000                   250                 \$7500
\$40                      \$26                     \$3000                   214                 \$8600
\$50                      \$29                     \$3000                   143                 \$7150

3.       If Christopher were selling 3 different types of baskets as shown in the table above, the sales would be across all types in an
unknown ratio. Therefore the only information he can deduce from the calculations for breakeven is that the number of baskets
needed to be sold to breakeven would be somewhere between 143 and 250. The range of the number of baskets that is needed to
be sold to breakeven is large and would therefore not be a good indicator to Christopher of the feasibility of his concept plan, ie if
143 baskets were sold of the\$30 type, Christopher would not breakeven.

4.       Choosing the Regular basket with a sale price of \$40, the sales income achieved to break-even is \$8600 as calculated in Q2.

5.       To calculate the breakeven number for the Regular basket with Sale price of \$40:

0                0- 3000                                                         -\$3000
100                100x40-(100x26+3000)                                            -\$1600
200                200x40-(200x26+3000)                                             -\$200
300                300x40-(300x26+3000)                                             \$1200
400                400x40-(400x26+3000)                                             \$2600
500                500x40-(500x26+3000)                                             \$4000

(This table could alternatively be calculated using a spreadsheet)

SSABSA Support Materials: 1e2a3fe2-710f-4ff6-b80b-3e0e85c65d96.doc, last updated 7 January 2003                         page 1 of 5

10000
8000
6000
Profit/Loss
4000
2000
0
-2000 0       100      200    300     400      500   600   700      800

-4000

6.         For this scenario the breakeven number was calculated at 214 gift baskets, and from the graph, it can be seen that the profit
line cuts the horizontal axis just above 200 baskets. This implies that Christopher needs to sell on average 4 baskets per week
across the year. Due to peak times throughout the year eg Easter, Mother’s Day and Christmas, when he would expect increased
sales, which would compensate for any slower sales weeks, Christopher believes that it is reasonable to assume that he could
exceed this number of baskets and therefore make a profit.
Therefore the breakeven number calculated is a very achievable and realistic value.

7.        Christopher predicts that he will sell 800 gift baskets in his first year, which represents on average, 15 sales per week.
Profit = approximately \$8200 (from the graph).           Algebraic answer is \$8200 as a check.

8.        Christopher’s taxable income is now \$25000 + \$8200 = \$33200. This includes his part time wage from working in the fruit

Taxable income = \$33200
Tax on \$20000 = \$ 2380
Tax on \$13200 = \$ 3960
Medicare levy = \$ 498
Total tax payable= \$ 6838

9.        Christopher’s net income = \$33200 - \$6838
= \$26362

10.       The Regular basket is currently priced at \$40, and there are approximately 800 sales of these baskets in a year. As shown this
produces an income of \$26362 annually for Christopher (including his part time income from working in the fruit and vegetable
shop).
Assuming the variable and fixed costs remain the same, the effect of increasing and decreasing the sale price effects the break
even number as shown in Table 1.

TABLE 1                                      VAR COSTS        \$       26.00
FIXED COSTS      \$     3,000.00

SALE PRICE              B/EVEN NO.

\$         40.00                   214.00
\$         30.00                   750.00
\$         45.00                   157.00
\$         60.00                   88.00

SSABSA Support Materials: 1e2a3fe2-710f-4ff6-b80b-3e0e85c65d96.doc, last updated 7 January 2003                        page 2 of 5
As can be seen from the table above, as the sale price increases, the breakeven number decreases, eg if Christopher sold the
baskets at \$30 he would need to sell 750 of them in a year to breakeven. If he sold the baskets at \$60, he would only have to sell
88 baskets in the year to breakeven. This effect of the increased sales price on the breakeven and therefore profit, is well
illustrated in the graph shown below:

\$30,000.00

\$25,000.00

\$20,000.00
Sale price \$40

\$15,000.00                                                                                                Sale Price \$30
Profit \$

Sale Price \$45
Sale Price \$60
\$10,000.00

\$5,000.00

\$-
100

200

300

400

500

600

700

800
0

-\$5,000.00
Number of Sales

The breakeven number clearly decreases as the sales increases. This can be seen from the graph by comparing the horizontal axis
intercept of each line. As the sale price increases, the slope of the line increases, indicating that the marginal income per unit has
increased. The fixed costs are the same for each scenario, shown by each line having the same vertical axis intercept. Calculating
the breakeven number is not he only factor that Christopher needs to consider.

If the sales price is increased, the sales profit also increases. The effect of the sales price on the sales profit and therefore
Christopher’s income can be seen in Table 2.

TABLE 2                  VAR COSTS              \$           26.00
FIXED COSTS            \$         3,000.00

SALE PRICE               B/EVEN NO.        NO OF BASKETS                PROFIT                TAXABLE INC        TAX PAYABLE          NET INCOME
SOLD
\$              40.00            214.00                   800           \$    8,200.00          \$    33,200.00         \$    6,838.00    \$       26,362.00
\$              30.00            750.00                   800           \$      200.00          \$    25,200.00         \$    4318.00     \$       20882.00
\$              45.00            157.00                   800           \$    12,200.00         \$    37,200.00         \$    8,098.00    \$       29,102.00
\$              60.00            88.00                    800           \$    24,200.00         \$    49,200.00         \$   11,878.00    \$       37,322.00

If the predicted sales volume of 800 baskets were maintained in a year, and an increase of \$5 to the sales price of \$45 is made, the
profit is increased from \$8200 to \$12200, and hence, Christopher’s net income increases to \$29102. If the sales price is increased
to \$60, the profit increases to \$24,200, almost 3 times the profit, and Christopher’s income increases to \$37322, a jump of nearly
\$11,000. If the sales price is decreased to \$30, the profit decreases from \$8200 to only \$200 with 800 sales made, and
Christopher’s net income is reduced to \$20882.

However, realistically, when the sale price is increased to any degree, the number of sales is likely to be reduced, because a
proportion of people will perceive the gift basket as being too expensive. If the sale price is decreased the number of sales made
is likely to increase, as the gift basket becomes more affordable and is seen as a bargain by more people. Therefore it is
unrealistic to assume 800 sales a year can be maintained if the sales price is changed. With the various sale prices being
investigated, Christopher predicts an adjustment in the number of sales made in a year as shown in table 3. This table then shows
the effect of the increase and decrease in sales on his net income.

SSABSA Support Materials: 1e2a3fe2-710f-4ff6-b80b-3e0e85c65d96.doc, last updated 7 January 2003                                                    page 3 of 5
TABLE 3
PREDICTED
SALE PRICE        B/EVEN NO.        NO OF BASKETS            PROFIT            TAXABLE INC       TAX PAYABLE           NET INC
SOLD
\$       40.00            214.00            800          \$     8,200.00         \$   33,200.00      \$     6,838.00    \$     26,362.00
\$       30.00            750.00            1000         \$     1,000.00         \$   26,000.00      \$     4,570.00    \$     21,430.00
\$       45.00            157.00            800          \$   12,200.00          \$   37,200.00      \$     8,098.00    \$     29,102.00
\$       60.00            88.00             400          \$   10,600.00          \$   35,600.00      \$     7,594.00    \$     28,006.00

If 800 sales were predicted for the original sale price of \$40, then an increase in this number of sales would be expected if the
sale price were reduced to \$30. Christopher predicts that 1000 may be sold at this price. Even with this increase in sales,
Christopher’s net income of \$21430 is still less than in the original scenario, where his income was \$26362.
Christopher predicts that if the sales price is increased by only \$5, it may not affect the sales over the course of the year. The
effect of this increase in sales price, increases Christopher’s income by \$2740 to \$29102.

Christopher considers some other sales volume scenarios as seen in table 4 below:

TABLE 4
PREDICTED
SALE PRICE        B/EVEN NO.         NO OF BASKETS             PROFIT            TAXABLE INC         TAX PAYABLE        NET INC
SOLD
\$       40.00           214.00               800          \$    8,200.00          \$   33,200.00   \$       6,838.00   \$      26,362.00
\$       30.00           750.00              1200          \$    1,800.00          \$   26,800.00   \$       4,822.00   \$      21,978.00
\$       45.00           157.00               700          \$   10,300.00          \$   35,300.00   \$       7,499.50   \$      27,800.50
\$       60.00           88.00                300          \$    7,200.00          \$   32,200.00   \$       6,523.00   \$      25,677.00

If the sale price is reduced to \$30 and the number of basket sold increases to 1200 a year, this still does not increase Christopher
income from the original scenario.
If the increase in price to \$45 did affect the number of sales made and this was reduced to 700 a year, the overall effect on
Christopher’s income is still an increase, but only by \$1438.
If the increase in sale price to \$60 were to affect the number of sales so that they were reduced to 300 a year, this would be
enough to reduce Christopher’s net income by \$685.

In table 5 below, Christopher has worked out the number of baskets to sell at each sale price so that his net income remains
similar.

TABLE 5
PREDICTED
SALE PRICE        B/EVEN NO.         NO OF BASKETS             PROFIT            TAXABLE INC     TAX PAYABLE            NET INC
SOLD
\$       40.00           214.00               800          \$    8,200.00          \$   33,200.00   \$      6,838.00    \$      26,362.00
\$       30.00           750.00              2800          \$    8,200.00          \$   33,200.00   \$      6,838.00    \$      26,362.00
\$       45.00           157.00               589          \$    8,191.00          \$   33,191.00   \$      6,835.17    \$      26,355.84
\$       60.00           88.00                330          \$    8,220.00          \$   33,220.00   \$      6,844.30    \$      26,375.70

2800 baskets would need to be sold at \$30 each if Christopher was to retain his income from the original scenario with sale price
of \$40 and 800 sales made. This would equate to selling around 54 gift baskets a week, which would be difficult to maintain in
the given situation even during special events such as mother’s day etc. this is mainly due to the fact that the sales price of \$30 is
only slightly higher than the variable costs of \$26, and the fixed costs still need to be covered. This remains an inappropriate sales
price to consider for the gift baskets unless the variable costs were reduced and hence the contents of the basket would be similar
With a slight increase in the sales price to \$45, Christopher would only need to sell 589, or slightly more than 10 per week to
retain the same net income, which would be quite achievable.
At \$60, 330 gift baskets would need to be sold to maintain the income received in the original scenario.

SSABSA Support Materials: 1e2a3fe2-710f-4ff6-b80b-3e0e85c65d96.doc, last updated 7 January 2003                                       page 4 of 5
Conclusions

From all the scenarios investigated, Christopher believes that if he wished to increase his income, he could easily increase the
price to \$45, without affecting the number of sales. To decrease the sale price to \$30 without changing the content of the basket
would most likely decrease his income as the volume of baskets needed to be sold in a year is very high and most likely not
achievable.
If Christopher were to increase the sale price to \$60, the number of sales is likely to decrease, as this is an increase of 50% in sale
price. Although 330 sales in a year may be achievable at this price, it may be more dependent on the peak times such as
Christmas, Easter and Mother’s Day, and may not produce a steady income for Christopher.
By considering all the scenarios, Christopher decides that \$40 or \$45 is an appropriate price for the Regular gift baskets, so that
he can maintain steady sales and income. However, this sales scenario does not take into account a number of factors:
The number of sales made in a year is not likely to be consistent either throughout a year or over a few years. If advertising were
to be used, this could increase the number of sales, but would also increase the fixed costs. If the baskets are of a good quality,
then word of mouth may increase sales and customers would return. On the other had if the baskets were not of good quality and
the service was poor, this could affect sales negatively.
Seasonal variations in fruits available may have an affect on the sales at certain times of the year.
The variable costs are not likely to be consistent throughout the year, as the price of fruit can vary with seasons and availability.
An increase in the variable costs would increase the breakeven price, and decrease the profit and Christopher’s income.
An increase in the fixed costs would also have a decreasing affect on Christopher’s income, ie an increase in the shop rental,
The figures calculated does not take into account the GST paid for good and received in sales. As there is no GST payable on
fresh fruit, the GST payable by Christopher is likely to be less than what is received in sales and therefore a proportion of the
profit would be payable to the taxation department.

Some other areas Christopher could investigate to help plan his business concept would be to look at the effects of increases or
decreases in the fixed and variable costs. Christopher was setting up this business within the established fruit and vegetable
business, so there was no inclusion of the cost of staffing this exercise. This would add an extra complexity and would certainly
add to the fixed costs quite considerably. By investigating these areas Christopher could predict and plan for any changes to his
profit and hence his net income.

(Appendices would include the spreadsheets and formulae used to calculate the tables shown in the report)

SSABSA Support Materials: 1e2a3fe2-710f-4ff6-b80b-3e0e85c65d96.doc, last updated 7 January 2003                         page 5 of 5

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